Social Choice Set Research Articles

Optimizing Over Serial Dictatorships

Motivated by the success of the serial dictatorship mechanism in social choice settings, we explore its usefulness in tackling various combinatorial optimization problems. We do so by considering an abstract model, in which a set of agents are asked to act in a particular ordering, called the action sequence. Each agent acts in a way that gives her the maximum possible value, given the actions of the agents who preceded her in the action sequence. Our goal is to compute action sequences that yield approximately optimal total value to the agents (a.k.a., social welfare). We assume query access to the value vi(S)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$v_i(S)$$\\end{document} that the agent i gets when she acts after the agents in the ordered set S. We establish tight bounds on the social welfare that can be achieved using polynomially many queries. Even though these bounds show a marginally sublinear approximation of optimal social welfare in general, excellent approximations can be obtained when the valuations stem from an underlying combinatorial domain. Indicatively, when the valuations are defined using bipartite matchings, arborescences in directed graphs, and satisfiability of Boolean expressions, simple query-efficient algorithms yield 2-approximations. We discuss issues related to truthfulness and show how some of our algorithms can be implemented truthfully using VCG-like payments. Finally, we introduce and study the price of serial dictatorship, a notion that provides an optimistic measure of the quality of combinatorial optimization solutions generated by action sequences.

Improved Metric Distortion via Threshold Approvals

We consider a social choice setting in which agents and alternatives are represented by points in a metric space, and the cost of an agent for an alternative is the distance between the corresponding points in the space. The goal is to choose a single alternative to (approximately) minimize the social cost (cost of all agents) or the maximum cost of any agent, when only limited information about the preferences of the agents is given. Previous work has shown that the best possible distortion one can hope to achieve is 3 when access to the ordinal preferences of the agents is given, even when the distances between alternatives in the metric space are known. We improve upon this bound of 3 by designing deterministic mechanisms that exploit a bit of cardinal information. We show that it is possible to achieve distortion 1+sqrt(2) by using the ordinal preferences of the agents, the distances between alternatives, and a threshold approval set per agent that contains all alternatives for whom her cost is within an appropriately chosen factor of her cost for her most-preferred alternative. We show that this bound is the best possible for any deterministic mechanism in general metric spaces, and also provide improved bounds for the fundamental case of a line metric.

The distortion of distributed metric social choice

We consider a social choice setting with agents that are partitioned into disjoint groups, and have metric preferences over a set of alternatives. Our goal is to choose a single alternative aiming to optimize various objectives that are functions of the distances between agents and alternatives in the metric space, under the constraint that this choice must be made in a distributed way: The preferences of the agents within each group are first aggregated into a representative alternative for the group, and then these group representatives are aggregated into the final winner. Deciding the winner in such a way naturally leads to loss of efficiency, even when complete information about the metric space is available. We provide a series of (mostly tight) bounds on the distortion of distributed mechanisms for variations of well-known objectives, such as the (average) total cost and the maximum cost, and also for new objectives that are particularly appropriate for this distributed setting and have not been studied before.

Multi-Party Campaigning

We study a social choice setting of manipulation in elections and extend the usual model in two major ways: first, instead of considering a single manipulating agent, in our setting there are several, possibly competing ones; second, instead of evaluating an election after the first manipulative action, we allow several back-and-forth rounds to take place. We show that in certain situations, such as in elections with only a few candidates, optimal strategies for each of the manipulating agents can be computed efficiently. Our algorithmic results rely on formulating the problem of finding an optimal strategy as sentences of Presburger arithmetic that are short and only involve small coefficients, which we show is fixed-parameter tractable -- indeed, one of our contributions is a general result regarding fixed-parameter tractability of Presburger arithmetic that might be useful in other settings. Following our general theorem, we design quite general algorithms; in particular, we describe how to design efficient algorithms for various settings, including settings in which we model diffusion of opinions in a social network, complex budgeting schemes available to the manipulating agents, and various realistic restrictions on adversary actions.

Aggregation over Metric Spaces: Proposing and Voting in Elections, Budgeting, and Legislation

We present a unifying framework encompassing a plethora of social choice settings. Viewing each social choice setting as voting in a suitable metric space, we offer a general model of social choice over metric spaces, in which—similarly to the spatial model of elections—each voter specifies an ideal element of the metric space. The ideal element acts as a vote, where each voter prefers elements that are closer to her ideal element. But it also acts as a proposal, thus making all participants equal not only as voters but also as proposers. We consider Condorcet aggregation and a continuum of solution concepts, ranging from minimizing the sum of distances to minimizing the maximum distance. We study applications of our abstract model to various social choice settings, including single-winner elections, committee elections, participatory budgeting, and participatory legislation. For each setting, we compare each solution concept to known voting rules and study various properties of the resulting voting rules. Our framework provides expressive aggregation for a broad range of social choice settings while remaining simple for voters; and may enable a unified and integrated implementation for all these settings, as well as unified extensions such as sybil-resiliency, proxy voting, and deliberative decision making. We study applications of our abstract model to various social choice settings, including single-winner elections, committee elections, participatory budgeting, and participatory legislation. For each setting, we compare each solution concept to known voting rules and study various properties of the resulting voting rules. Our framework provides expressive aggregation for a broad range of social choice settings while remaining simple for voters; and may enable a unified and integrated implementation for all these settings, as well as unified extensions such as sybil-resiliency, proxy voting, and deliberative decision making.

Peeking behind the ordinal curtain: Improving distortion via cardinal queries

Aggregating the preferences of individuals into a collective decision is the core subject of study of social choice theory. In 2006, Procaccia and Rosenschein considered a utilitarian social choice setting, where the agents have explicit numerical values for the alternatives, yet they only report their linear orderings over them. To compare different aggregation mechanisms, Procaccia and Rosenschein introduced the notion of distortion, which quantifies the inefficiency of using only ordinal information when trying to maximize the social welfare, i.e., the sum of the underlying values of the agents for the chosen outcome. Since then, this research area has flourished and bounds on the distortion have been obtained for a wide variety of fundamental scenarios. However, the vast majority of the existing literature is focused on the case where nothing is known beyond the ordinal preferences of the agents over the alternatives. In this paper, we take a more expressive approach, and consider mechanisms that are allowed to further ask a few cardinal queries in order to gain partial access to the underlying values that the agents have for the alternatives. With this extra power, we design new deterministic mechanisms that achieve significantly improved distortion bounds and, in many cases, outperform the best-known randomized ordinal mechanisms. We paint an almost complete picture of the number of queries required by deterministic mechanisms to achieve specific distortion bounds.

Full Implementation under Ambiguity

This paper introduces the maxmin expected utility framework into the problem of fully implementing a social choice set as ambiguous equilibria. Our model incorporates the Bayesian framework and the Wald-type maxmin preferences as special cases and provides insights beyond the Bayesian implementation literature. We establish necessary and almost sufficient conditions for a social choice set to be fully implementable. Under the Wald-type maxmin preferences, we provide easy-to-check sufficient conditions for implementation. As applications, we implement the set of ambiguous Pareto-efficient and individually rational social choice functions, the maxmin core, the maxmin weak core, and the maxmin value. (JEL D71, D81, D82)

Justified Representation for Perpetual Voting

We study the social choice setting of perpetual voting where, based on voter preferences, decisions have to be taken over a finite horizon of consecutive points in time (e.g, days). We consider two complementary settings: a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Static</i> setting, in which voter preferences remain static over time, and a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Dynamic</i> setting, in which voter preferences may change over time. We adapt the well-established Justified Representation and Proportional Justified Representation axioms, commonly used in the social choice literature, to perpetual voting, resulting in two axiomatic variants for the static setting and four variants for the dynamic setting. We show that all of the axioms are always satisfiable, and that simple preference aggregation methods can be used to satisfy the axioms in both cases. We then conduct a large human study ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N=190$ </tex-math></inline-formula> ) aimed at identifying what potential voters (i.e., ordinary people) deem as desirable outcomes in simple perpetual voting settings. Our results show that approximately half of our participants consider different interpretations of fairness that correspond to our axiomatic framework. Taken jointly, our results can be used to help the research community identify appropriate aggregation methods to use in practice.

A foundation for Pareto optimality

Can an axiomatic justification be given for the requirement that society picks all and only Pareto optimal alternatives at each profile of individual preferences? Using the framework of fixed-agenda social choice theory, we present a characterization of the Pareto optimal social choice correspondence. We introduce a new independence condition, P-independence. When combined with three natural assumptions, P-independence leads to the conclusion that the social choice set and the Pareto optimal set are the same.

Closure under interim utility equivalence implies two-agent Bayesian implementation

We study the relationship between the two-agent implementation problem and the concept of interim efficiency of Holmström and Myerson (1983) in Bayesian environments with private values and independent types. We present a general property, called closure under interim utility equivalence, and show it is sufficient for the implementation of social choice functions. This condition, when combined with another property, called interim inseparability, is also sufficient for the implementation of essentially single-valued social choice sets. The characterization results are then examined in a variety of environments.

The value of travel time, noise pollution, recreation and biodiversity: A social choice valuation perspective

Environmental effects of transport projects have a weak position in Cost-Benefit Analysis (CBA) which might be rooted in the valuation approach adopted in the dominant style of CBA. This conventional valuation approach has been criticized for not valuing positive and negative impacts of transport projects in relation to each other and for not valuing such impacts in a public context, but in the context of private decisions. These critiques might be circumvented through valuing transport projects in a social choice context in which overall burdens and benefits of proposed transport projects are considered together in a public context. We investigate the extent to which a social choice valuation approach produces different outcomes than a conventional valuation approach. We conducted four social choice valuation experiments in which respondents were asked to choose between alternatives for a new road, trading off travel time and three environmental impacts (noise, recreation and biodiversity). Our findings suggest that, under social choice valuation, individuals assign substantially more value to environmental impacts than travel time as compared to conventional valuation studies. Moreover, in a social choice setting, respondents assigned monetary values to impacts that are not (or only qualitatively) considered in conventional CBAs of transport projects.

New Approximations for Coalitional Manipulation in Scoring Rules

We study the problem of coalitional manipulation---where k manipulators try to manipulate an election on m candidates---for any scoring rule, with focus on the Borda protocol. We do so in both the weighted and unweighted settings. For these problems, recent approximation approaches have tried to minimize k, the number of manipulators needed to make some preferred candidate p win (thus assuming that the number of manipulators is not limited in advance). In contrast, we focus on minimizing the score margin of p which is the difference between the maximum score of a candidate and the score of p. We provide algorithms that approximate the optimum score margin, which are applicable to any scoring rule. For the specific case of the Borda protocol in the unweighted setting, our algorithm provides a superior approximation factor for lower values of k.Our methods are novel and adapt techniques from multiprocessor scheduling by carefully rounding an exponentially-large configuration linear program that is solved by using the ellipsoid method with an efficient separation oracle. We believe that such methods could be beneficial in other social choice settings as well.

Group Recommendations: Axioms, Impossibilities, and Random Walks

We introduce an axiomatic approach to group recommendations, in line of previous work on the axiomatic treatment of trust-based recommendation systems, ranking systems, and other foundational work on the axiomatic approach to internet mechanisms in social choice settings. In group recommendations we wish to recommend to a group of agents, consisting of both opinionated and undecided members, a joint choice that would be acceptable to them. Such a system has many applications, such as choosing a movie or a restaurant to go to with a group of friends, recommending games for online game players, & other communal activities. Our method utilizes a given social graph to extract information on the undecided, relying on the agents influencing them. We first show that a set of fairly natural desired requirements (a.k.a axioms) leads to an impossibility, rendering mutual satisfaction of them unreachable. However, we also show a modified set of axioms that fully axiomatize a group variant of the random-walk recommendation system, expanding a previous result from the individual recommendation case.

Social Networks and Choice Set Formation in Discrete Choice Models

The discrete choice literature has evolved from the analysis of a choice of a single item from a fixed choice set to the incorporation of a vast array of more complex representations of preferences and choice set formation processes into choice models. Modern discrete choice models include rich specifications of heterogeneity, multi-stage processing for choice set determination, dynamics, and other elements. However, discrete choice models still largely represent socially isolated choice processes —individuals are not affected by the preferences of choices of other individuals. There is a developing literature on the impact of social networks on preferences or the utility function in a random utility model but little examination of such processes for choice set formation. There is also emerging evidence in the marketplace of the influence of friends on choice sets and choices. In this paper we develop discrete choice models that incorporate formal social network structures into the choice set formation process in a two-stage random utility framework. We assess models where peers may affect not only the alternatives that individuals consider or include in their choice sets, but also consumption choices. We explore the properties of our models and evaluate the extent of “errors” in assessment of preferences, economic welfare measures and market shares if network effects are present, but are not accounted for in the econometric model. Our results shed light on the importance of the evaluation of peer or network effects on inclusion/exclusion of alternatives in a random utility choice framework.

Optimal Voting Rules

We study dominant strategy incentive compatible (DIC) and deterministic mechanisms in a social choice setting with several alternatives. The agents are privately informed about their preferences, and have single-crossing utility functions. Monetary transfers are not feasible. We use an equivalence between deterministic, DIC mechanisms and generalized median voter schemes to construct the constrained-efficient, optimal mechanism for an utilitarian planner. Optimal schemes for other welfare criteria such as, say, a Rawlsian maximin can be analogously obtained.

SOCIAL CHOICE AND TIME CONSISTENCY WITH LOW-PROBABILITY EVENTS

A key result in macroeconomics is the “time inconsistency of short-run optimal plans.” It is argued that inflationary bias results if central bankers do not precommit to a monetary policy rule. The macro literature, however, does not address the way in which board members arrive at the “optimal choice of inflation rate.” That is a matter of a micro subfield called social choice. If we consider that on any board, members have different priors about the states of nature for the economy, but they all receive the same signal before deciding, then they will have different posterior probabilities for the states, even if they have many data, if one state has a low probability of occurring, such as an unlikely catastrophic-risk event. Thus, it is not clear what the optimal plan is. Therefore, discretion rather than rules may be the optimal plan in social choice settings.

Implementability under monotonic transformations in differences

In a social choice setting with quasilinear preferences and monetary transfers, a domain D of admissible valuations is called a monotonicity domain if every 2-cycle monotone allocation rule is truthfully implementable (in dominant strategies). D is called a revenue equivalence domain if every implementable allocation rule satisfies the revenue equivalence property. We introduce the notions of monotonic transformations in differences, which can be interpreted as extensions of Maskin's monotonic transformations to quasilinear environments, and show that if D admits these transformations then it is a monotonicity and revenue equivalence domain. Our proofs are elementary and do not rely on strenuous additional machinery. We illustrate monotonic transformations in differences for settings with finite and infinite allocation sets.

Ex-Post Equilibrium and VCG Mechanisms

Consider an abstract social choice setting with incomplete information, where the number of alternatives is large. Albeit natural, implementing VCG mechanisms is infeasible due to the prohibitive communication constraints. However, if players restrict attention to a subset of the alternatives, feasibility may be recovered. This article characterizes the class of subsets that induce an ex-post equilibrium in the original game. It turns out that a crucial condition for such subsets to exist is the availability of a type-independent optimal social alternative for each player. We further analyze the welfare implications of these restrictions. This work follows that of Holzman et al. [2004] and Holzman and Monderer [2004] where similar analysis is done for combinatorial auctions.

A sufficient condition for two-person ex post implementation in a general environment

This paper establishes sufficient conditions for two-person ex post implementation of a social choice set in a general environment. A single social choice function is ex post implementable if it satisfies ex post incentive compatibility (EPIC) and ex post monotonicity no veto (EMNV) conditions. A general social choice set is ex post implementable if it satisfies another condition, intersection property (IP), in addition to the above.

The computational complexity of random serial dictatorship

In social choice settings with linear preferences, random dictatorship is known to be the only social decision scheme satisfying strategyproofness and ex post efficiency. When also allowing indifferences, random serial dictatorship (RSD) is a well-known generalization of random dictatorship that retains both properties. RSD has been particularly successful in the special domain of random assignment where indifferences are unavoidable. While executing RSD is obviously feasible, we show that computing the resulting probabilities is #P-complete, and thus intractable, both in the context of voting and assignment.